On a Problem in the Theory of Diffusion Processes

In the paper some Markov processes associated with diffusion processes are discussed, A diffusion process $x_t$ defined on $l$-dimensional Euclidean space $E^l$ is considered only at moments when its trajectory belongs to a given set $S$ (a new time is introduced which changes only when the process is in $S$). If $S$ is a domain with differentiable boundary, the generator $\tilde{\mathfrak{A}}$ of the new process $y_t$ is the same as for $x_t$ at all interior points of $S$. On the boundary of $S$ non-classical boundary conditions are obtained. These boundary conditions are described in Theorem 1. If $S$ is an $(l-1)$-dimensional surface, we obtain on $S$ a discontinuous process of Cauchy type. The generator of this process is investigated in Theorem 2.